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OGS
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A failure index dependent permeability model [44].
\mathbf{k} = \mathbf{k}_0+ H(f-1) k_\text{r} \mathrm{e}^{b f}\mathbf{I}
where \mathbf{k}_0 is the intrinsic permeability of the undamaged material, H is the Heaviside step function, f is the failure index, k_\text{r} is a reference permeability, b is a fitting parameter. k_\text{r} and b can be calibrated by experimental data.
The failure index f is calculated from the Mohr Coulomb failure criterion comparing an acting shear stress for the shear dominated failure. The tensile failure is governed by an input parameter of tensile_strength_parameter .
The Mohr Coulomb failure criterion [25] takes the form
\tau(\sigma)=c-\sigma \mathrm{tan} \phi
with \tau the shear stress, c the cohesion, \sigma the normal stress, and \phi the internal friction angle.
The failure index of the Mohr Coulomb model is calculated by
f_{MC}=\frac{|\tau_m| }{\cos(\phi)\tau(\sigma_m)}
with \tau_m=(\sigma_3-\sigma_1)/2 and \sigma_m=(\sigma_1+\sigma_3)/2, where \sigma_1 and \sigma_3 are the minimum and maximum shear stress, respectively.
The tensile failure index is calculated by
f_{t} = \sigma_m / \sigma^t_{max}
with, 0 < \sigma^t_{max} < c \tan(\phi) , a parameter of tensile strength for the cutting of the apex of the Mohr Coulomb model.
The tensile stress status is determined by a condition of \sigma_m> \sigma^t_{max}. The failure index is then calculated by
f = \begin{cases} f_{MC}, & \sigma_{m} \leq \sigma^t_{max}\\ max(f_{MC}, f_t), & \sigma_{m} > \sigma^t_{max}\\ \end{cases}
The computed permeability components are restricted with an upper bound, i.e. \mathbf{k}:=k_{ij} < k_{max}.
If \mathbf{k}_0 is orthogonal, i.e input two or three numbers for its diagonal entries, a coordinate system rotation of \mathbf{k} is possible if it is needed.
Note: the conventional mechanics notations are used, which mean that tensile stress is positive.
No additional info.
1.12.0